A090771 - OEIS

%I #61 Nov 23 2024 05:44:40

%S 1,9,11,19,21,29,31,39,41,49,51,59,61,69,71,79,81,89,91,99,101,109,

%T 111,119,121,129,131,139,141,149,151,159,161,169,171,179,181,189,191,

%U 199,201,209,211,219,221,229,231,239,241,249,251,259,261,269,271,279,281

%N Numbers that are congruent to {1, 9} mod 10.

%C Cf. property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n + (h-4)*(-1)^n - h)/4 (h, n natural numbers), therefore ((2*h*n + (h-4)*(-1)^n - h)/4)^2-1 == 0 (mod h); in this case, a(n)^2 - 1 == 0 (mod 10). - _Bruno Berselli_, Nov 17 2010

%H Reinhard Zumkeller, <a href="/A090771/b090771.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).

%F a(n) = sqrt(40*A057569(n) + 1). - _Gary Detlefs_, Feb 22 2010

%F From _Bruno Berselli_, Sep 16 2010 - Nov 17 2010: (Start)

%F   G.f.: x*(1 + 8*x + x^2)/((1 + x)*(1 - x)^2).

%F   a(n) = (10*n + 3*(-1)^n - 5)/2.

%F   a(n) = -a(-n + 1) = a(n-1) + a(n-2) - a(n-3) = a(n-2) + 10.

%F   a(n) = 10*A000217(n-1) + 1 - 2*Sum_{i=1..n-1} a(i) for n > 1. (End)

%F a(n) = 10*n - a(n-1) - 10 (with a(1) = 1). - _Vincenzo Librandi_, Nov 16 2010

%F a(n) = sqrt(10*A132356(n-1) + 1). - _Ivan N. Ianakiev_, Nov 09 2012

%F Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/10)*cot(Pi/10) = A000796 * A019970 / 10 = sqrt(5 + 2*sqrt(5))*Pi/10. - _Amiram Eldar_, Dec 04 2021

%F E.g.f.: 1 + ((10*x - 5)*exp(x) + 3*exp(-x))/2. - _David Lovler_, Sep 03 2022

%F From _Amiram Eldar_, Nov 23 2024: (Start)

%F Product_{n>=1} (1 - (-1)^n/a(n)) = sqrt(phi+2) (A188593).

%F Product_{n>=2} (1 + (-1)^n/a(n)) = Pi*phi/5 = A094888/10. (End)

%t Flatten[Table[10n - {9, 1}, {n, 30}]] (* _Alonso del Arte_, Sep 02 2014 *)

%t LinearRecurrence[{1,1,-1},{1,9,11},60] (* _Harvey P. Dale_, Jul 05 2020 *)

%o (Haskell)

%o a090771 n = a090771_list !! (n-1)

%o a090771_list = 1 : 9 : map (+ 10) a090771_list

%o -- _Reinhard Zumkeller_, Jan 07 2012

%o (PARI) a(n)=n\2*10-(-1)^n \\ _Charles R Greathouse IV_, Sep 24 2015

%Y Cf. A000217, A000796, A090298, A005408, A047209, A007310, A019970, A047336, A047522, A091998, A094888, A113801, A175886, A175887, A188593.

%Y Cf. A056020 (n = 1 or 8 mod 9), A175885 (n = 1 or 10 mod 11).

%Y Cf. A045468 (primes), A195142 (partial sums).

%K nonn,easy

%O 1,2

%A _Giovanni Teofilatto_, Feb 07 2004

%E Edited and extended by _Ray Chandler_, Feb 10 2004